• Previous Article
    Penalisation of long treatment time and optimal control of a tumour growth model of Cahn–Hilliard type with singular potential
  • DCDS Home
  • This Issue
  • Next Article
    Non-autonomous stochastic evolution equations with nonlinear noise and nonlocal conditions governed by noncompact evolution families
doi: 10.3934/dcds.2020392

Schrödinger Equations with vanishing potentials involving Brezis-Kamin type problems

1. 

Departamento de Matemática, Universidade Federal de Sergipe, São Cristóvão-SE, 49100-000, Brazil

2. 

Departamento de Matematica y C. C., Universidad de Santiago de Chile, Casilla 307, Correo 2, Santiago, Chile

3. 

Unidade Acadêmica de Matemática, Universidade Federal de Campina Grande, Campina Grande 58429-900, Brazil

* Corresponding author

Received  December 2019 Revised  October 2020 Published  December 2020

Fund Project: The first author is partially supported by FAPITEC/CAPES and by CNPq - Universal.
The second author was partially supported by Proyecto código 042033CL, Dirección de Investigación, Científica y Tecnológica, DICYT.
The third author was partially supported by Proyecto código 041933UL POSTDOC, Dirección de Investigación, Científica y Tecnológica, DICYT.
The fourth author was partially supported by FONDECYT grant 1181125, 1161635, 1171691

We prove the existence of a bounded positive solution for the following stationary Schrödinger equation
$ \begin{equation*} -\Delta u+V(x)u = f(x,u),\,\,\, x\in\mathbb{R}^n,\,\, n\geq 3, \end{equation*} $
where
$ V $
is a vanishing potential and
$ f $
has a sublinear growth at the origin (for example if
$ f(x,u) $
is a concave function near the origen). For this purpose we use a Brezis-Kamin argument included in [6]. In addition, if
$ f $
has a superlinear growth at infinity, besides the first solution, we obtain a second solution. For this we introduce an auxiliar equation which is variational, however new difficulties appear when handling the compactness. For instance, our approach can be applied for nonlinearities of the type
$ \rho(x)f(u) $
where
$ f $
is a concave-convex function and
$ \rho $
satisfies the
$ \mathrm{(H)} $
property introduced in [6]. We also note that we do not impose any integrability assumptions on the function
$ \rho $
, which is imposed in most works.
Citation: Jose Anderson Cardoso, Patricio Cerda, Denilson Pereira, Pedro Ubilla. Schrödinger Equations with vanishing potentials involving Brezis-Kamin type problems. Discrete & Continuous Dynamical Systems - A, doi: 10.3934/dcds.2020392
References:
[1]

A. AmbrosettiH. Brezis and G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal., 122 (1994), 519-543.  doi: 10.1006/jfan.1994.1078.  Google Scholar

[2]

A. AmbrosettiV. Felli and A. Malchiodi, Ground states of nonlinear Schrödinger equations with potentials vanishing at infinity, J. Eur. Math. Soc., 7 (2005), 117-144.  doi: 10.4171/JEMS/24.  Google Scholar

[3]

A. BahrouniH. Ounaies and V. D. Rădulescu, Bound state solutions of sublinear Schrödinger equations with lack of compactness, RACSAM, 113 (2019), 1191-1210.  doi: 10.1007/s13398-018-0541-9.  Google Scholar

[4]

A. BahrouniH. Ounaies and V. D. Rădulescu, Infinitely many solutions for a class of sublinear Schrödinger equations with indefinite potentials, Proc. Roy. Soc. Edinburgh Sect. A, 145 (2015), 445-465.  doi: 10.1017/S0308210513001169.  Google Scholar

[5]

H. Brezis and L. Oswald, Remarks on sublinear elliptic equations, Nonlinear Analysis. Theory, Methods & Applications., 1 (1986), 55-64.  doi: 10.1016/0362-546X(86)90011-8.  Google Scholar

[6]

H. Brezis and S. Kamin, Sublinear elliptic equations in $\mathbb{R}^N$, Manuscripta Math., 74 (1992), 87-106.  doi: 10.1007/BF02567660.  Google Scholar

[7]

H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36 (1983), 437-477.  doi: 10.1002/cpa.3160360405.  Google Scholar

[8]

J. Chabrowski and J. M. B. do Ó, On semilinear elliptic equations involving concave and convex nonlinearities, Math. Nachr., 233/234 (2002), 55-76.  doi: 10.1002/1522-2616(200201)233:1<55::AID-MANA55>3.0.CO;2-R.  Google Scholar

[9]

D. G. de FigueiredoJ-P Gossez and P. Ubilla, Local superlinearity and sublinearity for indefinite semilinear elliptic problems, J. Funct. Anal., 199 (2003), 452-467.  doi: 10.1016/S0022-1236(02)00060-5.  Google Scholar

[10]

D. G. de FigueiredoJ-P Gossez and P. Ubilla, Multiplicity results for a family of semilinear elliptic problems under local superlinearity and sublinearity, J. Eur. Math. Soc., 8 (2006), 269-286.  doi: 10.4171/JEMS/52.  Google Scholar

[11]

F. Gazzola and A. Malchiodi, Some remark on the equation $-\Delta u = \lambda(1+u)^p$ for varying $\lambda, p$ and varying domains, Comm. Partial Differential Equations, 27 (2002), 809-845.  doi: 10.1081/PDE-120002875.  Google Scholar

[12]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, 1983. doi: 10.1007/978-3-642-61798-0.  Google Scholar

[13]

Q. Han and F. Lin, Elliptic Partial Differential Equations, Courant Lect. Notes Math., vol. 1, AMS, Providence, RI, 1997.  Google Scholar

[14]

T-S Hsu and H-L Lin, Four positive solutions of semilinear elliptic equations involving concave and convex nonlinearities in $\mathbb{R}^n$, J. Math. Anal. Appl., 365 (2010), 758-775.  doi: 10.1016/j.jmaa.2009.12.004.  Google Scholar

[15]

Z. Liu and Z-Q Wang, Schrödinger equations with concave and convex nonlinearities, Z. angew. Math. Phys., 56 (2005), 609-629.  doi: 10.1007/s00033-005-3115-6.  Google Scholar

[16]

M. H. Protter and H. F. Weinberger, Maximum Principle in Differential Equations, Prentice Hall, Englewoood Cliffs, New Jersey, 1967.  Google Scholar

[17]

T-F Wu, Multiple positive solutions for a class of concave-convex elliptic problems in $\mathbb{R}^n$ involving sign-changing weight, J. Funct. Anal., 258 (2010), 99-131.  doi: 10.1016/j.jfa.2009.08.005.  Google Scholar

show all references

References:
[1]

A. AmbrosettiH. Brezis and G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal., 122 (1994), 519-543.  doi: 10.1006/jfan.1994.1078.  Google Scholar

[2]

A. AmbrosettiV. Felli and A. Malchiodi, Ground states of nonlinear Schrödinger equations with potentials vanishing at infinity, J. Eur. Math. Soc., 7 (2005), 117-144.  doi: 10.4171/JEMS/24.  Google Scholar

[3]

A. BahrouniH. Ounaies and V. D. Rădulescu, Bound state solutions of sublinear Schrödinger equations with lack of compactness, RACSAM, 113 (2019), 1191-1210.  doi: 10.1007/s13398-018-0541-9.  Google Scholar

[4]

A. BahrouniH. Ounaies and V. D. Rădulescu, Infinitely many solutions for a class of sublinear Schrödinger equations with indefinite potentials, Proc. Roy. Soc. Edinburgh Sect. A, 145 (2015), 445-465.  doi: 10.1017/S0308210513001169.  Google Scholar

[5]

H. Brezis and L. Oswald, Remarks on sublinear elliptic equations, Nonlinear Analysis. Theory, Methods & Applications., 1 (1986), 55-64.  doi: 10.1016/0362-546X(86)90011-8.  Google Scholar

[6]

H. Brezis and S. Kamin, Sublinear elliptic equations in $\mathbb{R}^N$, Manuscripta Math., 74 (1992), 87-106.  doi: 10.1007/BF02567660.  Google Scholar

[7]

H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36 (1983), 437-477.  doi: 10.1002/cpa.3160360405.  Google Scholar

[8]

J. Chabrowski and J. M. B. do Ó, On semilinear elliptic equations involving concave and convex nonlinearities, Math. Nachr., 233/234 (2002), 55-76.  doi: 10.1002/1522-2616(200201)233:1<55::AID-MANA55>3.0.CO;2-R.  Google Scholar

[9]

D. G. de FigueiredoJ-P Gossez and P. Ubilla, Local superlinearity and sublinearity for indefinite semilinear elliptic problems, J. Funct. Anal., 199 (2003), 452-467.  doi: 10.1016/S0022-1236(02)00060-5.  Google Scholar

[10]

D. G. de FigueiredoJ-P Gossez and P. Ubilla, Multiplicity results for a family of semilinear elliptic problems under local superlinearity and sublinearity, J. Eur. Math. Soc., 8 (2006), 269-286.  doi: 10.4171/JEMS/52.  Google Scholar

[11]

F. Gazzola and A. Malchiodi, Some remark on the equation $-\Delta u = \lambda(1+u)^p$ for varying $\lambda, p$ and varying domains, Comm. Partial Differential Equations, 27 (2002), 809-845.  doi: 10.1081/PDE-120002875.  Google Scholar

[12]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, 1983. doi: 10.1007/978-3-642-61798-0.  Google Scholar

[13]

Q. Han and F. Lin, Elliptic Partial Differential Equations, Courant Lect. Notes Math., vol. 1, AMS, Providence, RI, 1997.  Google Scholar

[14]

T-S Hsu and H-L Lin, Four positive solutions of semilinear elliptic equations involving concave and convex nonlinearities in $\mathbb{R}^n$, J. Math. Anal. Appl., 365 (2010), 758-775.  doi: 10.1016/j.jmaa.2009.12.004.  Google Scholar

[15]

Z. Liu and Z-Q Wang, Schrödinger equations with concave and convex nonlinearities, Z. angew. Math. Phys., 56 (2005), 609-629.  doi: 10.1007/s00033-005-3115-6.  Google Scholar

[16]

M. H. Protter and H. F. Weinberger, Maximum Principle in Differential Equations, Prentice Hall, Englewoood Cliffs, New Jersey, 1967.  Google Scholar

[17]

T-F Wu, Multiple positive solutions for a class of concave-convex elliptic problems in $\mathbb{R}^n$ involving sign-changing weight, J. Funct. Anal., 258 (2010), 99-131.  doi: 10.1016/j.jfa.2009.08.005.  Google Scholar

[1]

Rim Bourguiba, Rosana Rodríguez-López. Existence results for fractional differential equations in presence of upper and lower solutions. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1723-1747. doi: 10.3934/dcdsb.2020180

[2]

Alessandro Fonda, Rodica Toader. A dynamical approach to lower and upper solutions for planar systems "To the memory of Massimo Tarallo". Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021012

[3]

Xiyou Cheng, Zhitao Zhang. Structure of positive solutions to a class of Schrödinger systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020461

[4]

Claudianor O. Alves, Rodrigo C. M. Nemer, Sergio H. Monari Soares. The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field. Communications on Pure & Applied Analysis, 2021, 20 (1) : 449-465. doi: 10.3934/cpaa.2020276

[5]

Alex H. Ardila, Mykael Cardoso. Blow-up solutions and strong instability of ground states for the inhomogeneous nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2021, 20 (1) : 101-119. doi: 10.3934/cpaa.2020259

[6]

Haoyu Li, Zhi-Qiang Wang. Multiple positive solutions for coupled Schrödinger equations with perturbations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020294

[7]

Riadh Chteoui, Abdulrahman F. Aljohani, Anouar Ben Mabrouk. Classification and simulation of chaotic behaviour of the solutions of a mixed nonlinear Schrödinger system. Electronic Research Archive, , () : -. doi: 10.3934/era.2021002

[8]

Lingyu Li, Jianfu Yang, Jinge Yang. Solutions to Chern-Simons-Schrödinger systems with external potential. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021008

[9]

Zedong Yang, Guotao Wang, Ravi P. Agarwal, Haiyong Xu. Existence and nonexistence of entire positive radial solutions for a class of Schrödinger elliptic systems involving a nonlinear operator. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020436

[10]

Serge Dumont, Olivier Goubet, Youcef Mammeri. Decay of solutions to one dimensional nonlinear Schrödinger equations with white noise dispersion. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020456

[11]

Chungen Liu, Huabo Zhang. Ground state and nodal solutions for fractional Schrödinger-maxwell-kirchhoff systems with pure critical growth nonlinearity. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020292

[12]

Juntao Sun, Tsung-fang Wu. The number of nodal solutions for the Schrödinger–Poisson system under the effect of the weight function. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021011

[13]

Norman Noguera, Ademir Pastor. Scattering of radial solutions for quadratic-type Schrödinger systems in dimension five. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021018

[14]

Oussama Landoulsi. Construction of a solitary wave solution of the nonlinear focusing schrödinger equation outside a strictly convex obstacle in the $ L^2 $-supercritical case. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 701-746. doi: 10.3934/dcds.2020298

[15]

Juliana Fernandes, Liliane Maia. Blow-up and bounded solutions for a semilinear parabolic problem in a saturable medium. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1297-1318. doi: 10.3934/dcds.2020318

[16]

Philippe Laurençot, Christoph Walker. Variational solutions to an evolution model for MEMS with heterogeneous dielectric properties. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 677-694. doi: 10.3934/dcdss.2020360

[17]

Xing-Bin Pan. Variational and operator methods for Maxwell-Stokes system. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3909-3955. doi: 10.3934/dcds.2020036

[18]

Kihoon Seong. Low regularity a priori estimates for the fourth order cubic nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5437-5473. doi: 10.3934/cpaa.2020247

[19]

José Luis López. A quantum approach to Keller-Segel dynamics via a dissipative nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020376

[20]

Van Duong Dinh. Random data theory for the cubic fourth-order nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020284

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (19)
  • HTML views (61)
  • Cited by (0)

[Back to Top]